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Solving Random Planted CSPs Below the n^{k/2} Threshold

Authors Arpon Basu , Jun-Ting Hsieh , Andrew D. Lin , Peter Manohar

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ACM Subject Classification
  • Mathematics of computing → Probabilistic algorithms
Keywords
  • Random CSPs
  • Sparse Learning Parity with Noise

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Abstract

We present a family of algorithms to solve random planted instances of any k-ary Boolean constraint satisfaction problem (CSP). A randomly planted instance of a Boolean CSP is generated by (1) choosing an arbitrary planted assignment x^*, and then (2) sampling constraints from a particular "planting distribution" designed so that x^* will satisfy every constraint. Given an n variable instance of a k-ary Boolean CSP with m constraints, our algorithm runs in time n^O(𝓁) for a choice of a parameter 𝓁, and succeeds in outputting a satisfying assignment if m ⩾ O(n)⋅(n/𝓁)^{k/2 - 1} log n. This generalizes the poly(n)-time algorithm of [Vitaly Feldman et al., 2015], the case of 𝓁 = O(1), to larger runtimes, and matches the constraint number vs. runtime trade-off established for refuting random CSPs by [Prasad Raghavendra et al., 2017].
Our algorithm is conceptually different from the recent algorithm of [Venkatesan Guruswami et al., 2023], which gave a poly(n)-time algorithm to solve semirandom CSPs with m ⩾ Õ(n^{k/2}) constraints by exploiting conditions that allow a basic SDP to recover the planted assignment x^* exactly. Instead, we forego certificates of uniqueness and recover x^* in two steps: we first use a degree-O(𝓁) Sum-of-Squares SDP to find some x̂ that is o(1)-close to x^*, and then we use a second rounding procedure to recover x^* from x̂.

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Arpon Basu, Jun-Ting Hsieh, Andrew D. Lin, and Peter Manohar. Solving Random Planted CSPs Below the n^{k/2} Threshold. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 23:1-23:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.23

Author Details

Arpon Basu
  • Princeton University, NJ, USA
Jun-Ting Hsieh
  • MIT, Cambridge, MA, USA
Andrew D. Lin
  • Princeton University, NJ, USA
Peter Manohar
  • The Institute for Advanced Study, Princeton, NJ, USA

Funding

  • Basu, Arpon: This material is based upon work supported by the Moskewicz Venture Forward Graduate Fellowship and NSF CAREER Award #2047933.
  • Hsieh, Jun-Ting: This material is based upon work supported by NSF CAREER Award #2047933.
  • Lin, Andrew D.: This material is based upon work supported by Princeton AI Lab Seed Grant.
  • Manohar, Peter: This material is based upon work supported by NSF Grant No. DMS-1926686

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